CHAPTER I An Overview 1. Introduction 1.1. Summary. This monograph is essentially the result of an unsuccess- ful attempt to give an updated account of Quillen's closed model categories, i.e. categories with three distinguished classes of maps (called weak equivalences, cofi- brations and fibrations) satisfying a few simple axioms which enable one to "do homotopy theory". That attempt however failed because (see 25.1), the deeper we got into the subject, and especially when we tried to deal with homotopy colimit and limit functors on arbitrary model categories, the more we realized that we did not really understand the role of the weak equivalences. In this introductory chapter we will therefore try to explain the problems we ran into and the solutions we came up with, and how all this determined the content and the layout of this monograph. 1.2. Organization of the chapter. After fixing some slightly unconven- tional terminology (in §2), we discuss the problems we encountered in dealing with the homotopy category of a model category (in §3) and with the homotopy colimit functors on a model category (in §4) and explain (in §5) how all this led to the cur- rent two-part monograph. In the first part we review some mostly known results on model categories, but do so whenever possible from the "homotopical" point of view which we develop in the second part, and in the second part we investigate what we call homotopical categories which are categories with only a single distinguished class of maps (called weak equivalences). The last section (§6) then contains some more details on this second part. 2. Slightly unconventional terminology In order to be able to give a reasonably clear formulation of the above men- tioned problems and their solutions, we slightly modify the customary meaning of the terms "model category" and "category" and introduce, just for use in this introductory chapter, the notion of a "category with weak equivalences" or "we- category". 2.1. Model categories. In [Qui67] Quillen introduced the notion of a model category, but then almost right away only concerned himself with the slightly more restricted but also more useful closed model categories which in [Qui69] he char- acterized by five simple axioms. However since then it has become clear that it is more convenient to restrict the definition even further by (i) strengthening his limit axiom, which requires the existence of all finite colimits and limits, by requiring the existence of all small colimits and limits, and 3
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